Commit 6292e58b authored by Leonard Guetta's avatar Leonard Guetta
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parent ea793336
......@@ -67,10 +67,12 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv
\]
where $H_k : \ho(\Ch) \to \Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$\nbd-th homology group.
\end{paragr}
\iffalse\begin{paragr}
In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.
Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.
\end{paragr}\fi
%% \begin{paragr}
%% In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.
%% Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.
%% \end{paragr}
\begin{remark}
The adjective ``singular'' is there to avoid future confusion with another homological invariant for $\oo$\nbd-categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of $\oo$-categories} ought to be simply called \emph{homology of $\oo$\nbd-categories} as it is the only ``correct'' definition of homology. This assertation will be justified later. \todo{Le faire !}
\end{remark}
......@@ -210,30 +212,30 @@ As we shall now see, when the $\oo$\nbd-categoru $C$ is \emph{free} the chain co
from the previous paragraph is an isomorphism.
\end{lemma}
\begin{proof}
\iffalse Let $G$ be an abelian group. For any $n \in \mathbb{N}$, we define an $n$-category $B^nG$ with:
\begin{itemize}
\item[-] $(B^nG)_{k}$ is a singleton set for every $k < n$,
\item[-] $(B^nG)_n = G$
\item[-] for all $x$ and $y$ in $G$ and $i<n$,
\[x \ast_i y := x +y.\]
\end{itemize}
It it straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.
%% Let $G$ be an abelian group. For any $n \in \mathbb{N}$, we define an $n$-category $B^nG$ with:
%% \begin{itemize}
%% \item[-] $(B^nG)_{k}$ is a singleton set for every $k < n$,
%% \item[-] $(B^nG)_n = G$
%% \item[-] for all $x$ and $y$ in $G$ and $i<n$,
%% \[x \ast_i y := x +y.\]
%% \end{itemize}
%% It it straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.
This defines a functor
\[
\begin{aligned}
B^n : \Ab &\to n\Cat\\
G &\mapsto B^nG,
\end{aligned}
\]
which is easily seen to be right adjoint to the functor
\[
\begin{aligned}
n\Cat &\to \Ab\\
X &\mapsto \lambda_n(X).
\end{aligned}
\]
\fi
%% This defines a functor
%% \[
%% \begin{aligned}
%% B^n : \Ab &\to n\Cat\\
%% G &\mapsto B^nG,
%% \end{aligned}
%% \]
%% which is easily seen to be right adjoint to the functor
%% \[
%% \begin{aligned}
%% n\Cat &\to \Ab\\
%% X &\mapsto \lambda_n(X).
%% \end{aligned}
%% \]
%%
Notice first that for any $\oo$\nbd-category $C$, we have $\lambda_n(\tau_{\leq n}^s(C))=\lambda_n(C)$. Suppose now that $C$ is free with basis $\Sigma=(\Sigma_n)_{n \in \mathbb{N}}$. Using Lemma \ref{lemma:adjlambdasusp} and Lemma \ref{lemma:freencattomonoid}, we obtain that for any abelian group $G$, we have
\begin{align*}
\Hom_{\Ab}(\lambda_n(C),G) &\simeq \Hom_{\Ab}(\lambda_n(\tau_{\leq n}^s(C)),G)\\
......@@ -574,11 +576,11 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins
\end{tikzcd}
\]
A throrough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following descritption of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$
\iffalse
\begin{tikzcd}
c_{\oo} : \Psh{\Delta} \ar[r,shift left] &\oo\Cat : N_{\oo}\ar[l,shift left]
\end{tikzcd}
\fi
%% \begin{tikzcd}
%% c_{\oo} : \Psh{\Delta} \ar[r,shift left] &\oo\Cat : N_{\oo}\ar[l,shift left]
%% \end{tikzcd}
with the abelianization functor, we obtain $2$-morphism
\[
\lambda c_{\oo} N_{\oo} \Rightarrow \lambda.
......@@ -706,8 +708,8 @@ Another consequence of the above counter-example is the following result, which
\]
which we know is impossible.
\end{proof}
\begin{paragr}
Even though triangle is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd-square
\begin{paragr}\label{paragr:defcancompmap}
Even though triangle \eqref{cmprisontrngle} is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd-square
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\Th}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
......@@ -732,20 +734,27 @@ Another consequence of the above counter-example is the following result, which
\ar[from=3-1,to=B,Rightarrow,"\pi",shorten <= 1em, shorten >= 1em]
\end{tikzcd}
\]
\iffalse \begin{equation}\label{trianglecomparisonmap}
\begin{tikzcd}
\sH^{\pol}\circ \gamma^{\folk} \ar[r,"\pi\ast\gamma^{\folk}",Rightarrow] \ar[rd,"\alpha^{\folk}\circ (\pi \ast \gamma^{\folk})"',Rightarrow] & \sH^{\sing}\circ \J \circ \gamma^{\folk} \ar[d,"\alpha^{\sing}\ast (\J \circ \gamma^{\folk})",Rightarrow]\\
&\gamma^{\Ch}\circ \lambda
\end{tikzcd}
\end{equation}
is commutative.
\fi
%% \begin{equation}\label{trianglecomparisonmap}
%% \begin{tikzcd}
%% \sH^{\pol}\circ \gamma^{\folk} \ar[r,"\pi\ast\gamma^{\folk}",Rightarrow] \ar[rd,"\alpha^{\folk}\circ (\pi \ast \gamma^{\folk})"',Rightarrow] & \sH^{\sing}\circ \J \circ \gamma^{\folk} \ar[d,"\alpha^{\sing}\ast (\J \circ \gamma^{\folk})",Rightarrow]\\
%% &\gamma^{\Ch}\circ \lambda
%% \end{tikzcd}
%% \end{equation}
%% is commutative.
Since $\J$ is nothing but the identity on objects, for any $\oo$\nbd-category $C$, the natural transformation $\pi$ yields a map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C),
\]
which we shall refer to as the \emph{canonical comparison map.}
\end{paragr}
\begin{remark}
When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\alpha^{\Ch}$ of the morphism of $\Ch$
\[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\]
induced by the co-unit of $c_{\oo} \dashv N_{\oo}$.
\end{remark}
% This motivates the following definition.
\begin{definition}
An $\oo$\nbd-category $C$ is said to be \emph{\good{}} when the canonical comparison map
......@@ -754,19 +763,35 @@ Another consequence of the above counter-example is the following result, which
\]
is an isomorphism of $\ho(\Ch)$.
\end{definition}
\begin{remark}
When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\alpha^{\Ch}$ of the morphism of $\Ch$
\[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\]
induced by the co-unit of $c_{\oo} \dashv N_{\oo}$.
\end{remark}
\begin{paragr}
Examples of \good{} $\oo$-categories will be presented later. The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd-categories. Note that if we think of $\oo\Cat$ as a model for the homotopy theory of spaces via Thomason equivalences (which is an informal way of stating Theorem \ref{thm:gagna}), then it follows from Proposition \ref{prop:polhmlgynotinvariant} that the polygraphic homology of an $\oo$\nbd-category, up to Thomason equivalence, is not well defined. With this perspective, polygraphic homology can be thought of a way to compute singular homology of \good{} $\oo$\nbd-category.
\begin{paragr}
The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd-categories. Examples of such $\oo$\nbd-categories will be presented later. Note that if we think of $\oo$\nbd-categories up to Thomason equivalences as spaces (which is an informal way of stating Theorem \ref{thm:gagna}), then it follows from Proposition \ref{prop:polhmlgynotinvariant} that polygraphic homology is not well defined. With this perspective, polygraphic homology can be thought of a way to compute singular homology of \good{} $\oo$\nbd-categories.
\end{paragr}
\section{A criterion to detect \good{} $\oo$\nbd-categories}
We shall now proceed to give an abstract criterion to find \good{} $\oo$-categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$-categories.
\begin{paragr}
Similarly to \ref{paragr:cmparisonmap}, the morphism of localizers
\[
(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})
\]
induces a morphism of op-prederivators
\[
\J : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th})
\]
such that the triangle
\[
\gamma^{\Th} = \J \circ \gamma^{\folk}
\]
is commutative. Moreover, all the constructions from \ref{paragr:defcancompmap} may be reproduced \emph{mutatis mutandis} at the level of op-prederivators. In particular, we obtain a $2$\nbd-morphism of op-prederivators
\[
\begin{tikzcd}
\Ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[rd,"\sH^{\pol}",""{name=A,below}] & \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch)\ar[from=2-1,to=A,"\pi",Rightarrow].
\end{tikzcd}
\]
\end{paragr}
\begin{paragr}\label{paragr:compcriterion}
Both the polygraphic homology
\[
......
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